In this 13th/14th-century copy of Cleomedes’ textbook on astronomy, typically referred to as “De motu circulari corporum caelestium,” is this illustration of Eratosthenes’ method for calculating the circumference of the earth. The outer oval is the signs of the zodiac in their zoomorphic/anthropomorphic forms, each labeled with its name. The sun is located in Cancer, in the upper right, illustrated by a head looking down and the label “ἥλιος.” Rays of light stream down on two cities. On the right is Syene; on the left is Alexandria (labeled accordingly). The city walls of each are adorned with a sundial. The gnomon on the dial in Syene is casts no shadow, since the sun shines down directly on the city. The gnomon in Alexandria would clearly cast a shadow, since the sunlight shines on the town at an angle. Between the two towns is a distance of 5,000 stadia, as noted by the “τὸ μεταξὺ στάδια Ε.” In the bottom is an illustration showing the path of the sun.

Cleomedes was a Greek astronomer active sometime in late antiquity (scholars don’t agree on when he lived). He is known primarily for his work, On the Circular Motions of the Celestial Bodies, a kind of textbook on astronomy. He preserves previous authors’ work, especially much of Posidonius’s work. He is also the earliest source for the story illustrated here about how Eratosthenes measured the circumference of the earth.

One recent sunny afternoon, I took a bunch of exercise balls with little sticks taped to them to the local grammar school where I met a class of second graders. As part of my war on the flat earth myth, I had encouraged their teacher to read Kathryn Lasky’s The Librarian Who Measured the Earth to them, and I had already come to class once to explain Eratosthenes’ method for measuring the earth’s circumference.

A picture book celebrating Eratosthenes. Not perfect but pretty darned good.

They seemed to get it, mostly. But I was left wishing for a more concrete, experiential way of showing them what Eratosthenes did. So I devised this hands-on exercise that they could do in groups of three.

A prototype model “earth” with gnomon. The final versions were slightly larger.

I used inflatable an exercise ball as our model “earth.” I taped pipe cleaners to them at two points on a line of latitude (a fabrication seam) as gnomons. I gave each group a tape measure. I explained that they were going to rotate their “earths” until one of the gnomon cast no shadow. Then, holding the ball still, they needed to measure the shadow cast by the other gnomon. They also needed to measure the height of this gnomon. Finally, they needed to measure the distance between the gnomons. I handed out a worksheet I had prepared so all they needed to do was fill in the first three columns on the table. I had them carry out the steps three times (one for each student). When they finished, they were to bring their sheets to me so I could calculate the circumference of their “earth.” Pretty basic instructions that even second graders can follow.

Worksheet for students to measure the circumference of their model earth.

They then came to me with their data. I plugged their numbers into a simple spreadsheet I had made (I confess, I cheated in so far as I used trigonometry to calculate the angle of the shadow cast by the gnomon). Their numbers were reasonably accurate (especially given the size of the ball and the uncertainty in the measurements).

Thirty minutes later, I had 33 second graders who not only knew that Eratosthenes had calculated the circumference of the earth, but could give a coherent account of how he did it. They eagerly took home their completed worksheets. Judging by the number of parents who have said something about it, they were able to explain to their parents what they had done and how.

For me, this is an important form of outreach, a way of “taking history of science to ‘them’.” Do I get any professional credit for it? Nope. Does it make the world a better place? Yep.

If you’re interested in more details, contact me. I’m happy to share.

Next up: I’m trying to convince the school to let me and the students use the flagpole as the gnomon for a permanent sundial.